Furthermore, a principal comodule algebra is the same as a faithfully-flat
Hopf–Galois extension, i.e., a Hopf–Galois extension \(A\)  of \(B\)  such that the tensor
product functor \((-)\otimes _B A\colon {M}{}\!\!_B \rightarrow {M}{}\!\!_A\)  both preserves and reflects
exact sequences. This gives an algebraic geometry flavour to the notion of a principal
comodule algebra, as it leads to the faithfully flat descent property. As observed by H.-J. Schneider, in one of the key results of Hopf–Galois theory [1]},
if \(H\)  admits an invariant integral (i.e., \(H\)  is coseparable) such as the Haar
measure on the coordinate algebra of a compact quantum group [2]},
then surjectivity of the canonical Galois map (REF ) implies its injectivity
as well as faithful-flatness of \(A\)  as a left \(B\) -module.
An explicit construction of a strong connection, in the more general situation of
extensions by coalgebras, is given in [3]}.
As a consequence, Hopf–Galois extensions given by typical Hopf algebras that
feature in noncommutative geometry, and in particular those found in the present text,
are automatically principal comodule algebras.
